We obtain an upper bound on the first Chern class and the Castelnuovo-Mumford regularity of an initialized rank 3 ACM bundle on a general hypersurface in P4. As a corollary, we prove that a general hypersurface in P4 of degree d≥ 4 does not support a rank 3 Ulrich bundle. We also make progress on the base case of a generic version of a conjecture by Buchweitz, Greuel and Schreyer. © 2022, The Author(s), under exclusive licence to Springer Nature B.V.